Chapter 6 - 6 Numerical Solutions of Ordinary Differential...

h =0.1h =0.05xyx(n)y(n)1.005.00001.005.00001.103.99001.054.44751.203.25461.103.97631.302.72361.153.57511.402.34511.203.23421.502.08011.252.94521.302.70091.352.49521.402.32261.452.17861.502.0592h =0.1h =0.05x(n)y(n)x(n)y(n)0.002.00000.002.00000.101.66000.051.81500.201.41720.101.65710.301.25410.151.52370.401.15640.201.41240.501.11220.251.32120.301.24820.351.19160.401.14990.451.12170.501.1056h =0.1h =0.05x(n)y(n)x(n)y(n)0.000.00000.000.00000.100.10050.050.05010.200.20300.100.10040.300.30980.150.15120.400.42340.200.20280.500.54700.250.25540.300.30950.350.36520.400.42300.450.48320.500.54656Numerical Solutions ofOrdinary Differential EquationsExercises 6.1All tables in this chapter were constructed in a spreadsheet program which does not support subscripts. Consequently,xnandynwill be indicated asx(n) andy(n), respectively.1.2.3.266

This preview
has intentionally blurred sections.
Sign up to view the full version.

This preview
has intentionally blurred sections.
Sign up to view the full version.

1.11.21.31.4x5101520yIMPROVEDh=0.1EULEREULERx(n)y(n)y(n)1.001.00001.00001.101.20001.24691.201.49381.66681.301.97112.64271.402.90608.7988Exercises 6.112. (a)(b)13. (a)Using the Euler method we obtainy(0.1)≈y1= 1.2.(b)Usingy= 4e2xwe see that the local truncation error isy(c)h22= 4e2c(0.1)22= 0.02e2c.Sincee2xis an increasing function,e2c≤e2(0.1)=e0.2for 0≤c≤0.1. Thus an upper bound for the localtruncation error is 0.02e0.2= 0.0244.(c)Sincey(0.1) =e0.2= 1.2214, the actual error isy(0.1)−y1= 0.0214, which is less than 0.0244.(d)Using the Euler method withh= 0.05 we obtainy(0.1)≈y2= 1.21.(e)The error in (d) is 1.2214−1.21 = 0.0114. With global truncation errorO(h), when the step size is halvedwe expect the error forh= 0.05 to be one-half the error whenh= 0.1. Comparing 0.0114 with 0.214 wesee that this is the case.14. (a)Using the improved Euler method we obtainy(0.1)≈y1= 1.22.(b)Usingy= 8e2xwe see that the local truncation error isy(c)h36= 8e2c(0.1)36= 0.001333e2c.Sincee2xis an increasing function,e2c≤e2(0.1)=e0.2for 0≤c≤0.1. Thus an upper bound for the localtruncation error is 0.001333e0.2= 0.001628.(c)Sincey(0.1) =e0.2= 1.221403, the actual error isy(0.1)−y1= 0.001403 which is less than 0.001628.(d)Using the improved Euler method withh= 0.05 we obtainy(0.1)≈y2= 1.221025.(e)The error in (d) is 1.221403−1.221025 = 0.000378. With global truncation errorO(h2), when the step sizeis halved we expect the error forh= 0.05 to be one-fourth the error forh= 0.1. Comparing 0.000378 with0.001403 we see that this is the case.

This is the end of the preview.
Sign up
to access the rest of the document.

What students are saying

As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran
Temple University Fox School of Business ‘17, Course Hero Intern

I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana
University of Pennsylvania ‘17, Course Hero Intern

The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.